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A quasilinear predator-prey model with indirect prey-taxis. (English) Zbl 1472.35218

Summary: This paper deals with a quasilinear predator-prey model with indirect prey-taxis \[ \begin{cases} u_t=\nabla \cdot (D(u)\nabla u)-\nabla \cdot (S(u)\nabla w)+rug(v)-uh(u), & (x,t)\in \Omega \times (0,\infty ),\\ w_t=d_w\Delta w- \mu w+\alpha v, & (x,t)\in \Omega \times (0,\infty ),\\ v_t=d_v\Delta v+f(v)-ug(v), & (x,t)\in \Omega \times (0,\infty ), \end{cases} \] under homogeneous Neumann boundary conditions in a smooth bounded domain \(\Omega \subset \mathbb{R}^n\), \(n\ge 1\), where \(d_w, d_v, \alpha, \mu, r>0\) and the functions \(g,h,f \in C^2([0,\infty ))\). The nonlinear diffusivity \(D\) and chemosensitivity \(S\) are supposed to satisfy \[ D(s)\ge a(s+1)^{-\gamma} \quad \text{and} \quad 0\le S(s)\le bs(s+1)^{\beta -1} \quad\text{for all}\quad s\ge 0, \] with \(a,b>0\) and \(\gamma, \beta\in \mathbb{R}\). Suppose that \(\gamma+\beta <1+\frac{1}{n}\) and \(\gamma <\frac{2}{n}\), it is proved that the problem has a unique global classical solution, which is uniformly bounded in time. In addition, we derive the asymptotic behavior of globally bounded solution in this system according to the different predation conditions.

MSC:

35K51 Initial-boundary value problems for second-order parabolic systems
35K59 Quasilinear parabolic equations
35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
92C17 Cell movement (chemotaxis, etc.)
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